used a Stimmgabelklangento generate tones that were ‘poor in over-tones’, found that sub-
jects judged the fourth to be slightly more ‘pleasant’than the tritone. Malmberg,^32 who
used tuning forks, found a more marked preference for the fourth over the tritone for judge-
ments of‘blending’,‘purity’, and ‘smoothness’. Pratt,^73 who used a Stern variator that may
have produced weak overtones, found that the fourth was judged to be more ‘pleasant’,
‘smoother’, and more ‘unitary’than the tritone. Brues,^74 using a Stern variator that pro-
duced weak energy at the first overtone, found the fourth, tritone, and fifth were similar
with respect to ‘unitariness’. Guthrie and Morrill^75 used a Stern variator that produced ‘very
faint traces of the third partial’and reported that the fourth was judged to be more ‘pleas-
ant’than the tritone and of equal ‘consonance’. Guernsey,^33 who used tuning forks,
reported that nonmusicians, amateur musicians, and professional musicians found the
fourth more ‘pleasant’and ‘smooth’than the tritone. Schellenberg and Trehub’s recent
experiments with nine-month-old babies are also relevant here.^76 When the upper pure
tone of a repeating harmonic interval was flattened by one-fourth of a semitone, infants
could detect the change if the interval was a fourth but not if it was a tritone. Their find-
ings indicate that fourths provided a more stable background against which changes in
tuning could be detected. All in all, these results indicate that the fourth, even when it is
composed of two pure tones, is often perceived as more con-sonant than the tritone.
Another challenge for roughness-based accounts of consonance arises when we compare
the consonance of pure-tone intervals and the consonance of complex-tone intervals. In
Figure 9.6D, Terhardt^13 plots consonance ratings for pure-tone and complex-tone intervals
on the same scale. In fact, Kameoka and Kuriyagawa’s psychoacoustic data and calculations
put them on different scales^35 (Figure 9.6F). Likewise, Plomp and Levelt use a dissonance
scale from one to zero for their pure-tone data (Figure 9.6C) and a dissonance scale from
six to zero for their complex-tone calculations (not shown).^51 Kameoka and Kuriyagawa’s
calculations predict that a minor second composed of two pure tones is more consonant
than the unison of two complex tones with the first six harmonics at isoamplitude^35
(Figure 9.6F). Intuitively, this notion is untenable; however, direct comparisons between
pure-tone intervals and complex-tone intervals have not been reported in the literature. We
synthesized a pure-tone minor second and the unison of two complex tones (with the
acoustic parameters specified by Kameoka and Kuriyagawa^35 ) and asked several of our stu-
dents to judge which of these two stimuli sounded more ‘consonant’. These and Huron’s
(personal communication) informal observations raise the possibility that a combination
of pitch height effects and loudness (shrillness), rather than or in addition to roughness,
accounts for Kameoka and Kuriyagaw’s predictions. For example, in the case of unison at
A 4 (Figure 9.6F), spectral energy extends all the way up to 2200 Hz (fifth harmonic) and
2640 Hz (sixth harmonic), so there are high-frequency components that are greater in sen-
sation level than the note F 0 s. At the same time,F(440 Hz) is higher than the highest F
associated with just-noticeable roughness when the root is at 2000 Hz (Figure 9.6A,
interquartile range for F~ 250–400 Hz^67 ).
In summary, the neural coding mechanisms that provide representations of roughness
form part of the neurobiological foundation for the theory of harmony in its vertical
dimension. However, our reappraisal of the psychoacoustic literature leads us to conclude
that the dependence of consonance on the absence of roughness is overstated. We believe
143